3. Consider a network (G, w). Let c ER, and let m: E(G) → R such that m(e) =w(e) + c for all e Є E(G).(a) Show that T is a minimum spanning tree of (G, w) if and only if it is a minimumspanning tree of (G,m).

Ans-3)(a) To show that T is a minimum spanning tree of (G, w) if and only if it is a minimum…

Answered: 3. Consider a network (G, w). Let c €…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Let G be the graph on vertices V1, V2, ..., V8 with the following adjacency matrix: /01000001 1 0 0 10000 100 100 1 000001 1 0 00 100 000 0001000 0001 0000 1 1 00000 i. Use the connectedness algorithm to find the connected component of v1. ii. Sketch all the graphs (up to isomorphism) on four vertices that are isomorphic to an induced subgraph of G (identifying a suitable induced subgraph in each case). iii. Sketch all the graphs (up to isomorphism) on four vertices that are not iso- morphic to an induced subgraph of G. iv. Find a graph on four vertices that is isomorphic to a subgraph, but not to an induced subgraph, of G. Explain your answer.
(6) 13. Let S = {a,b,c,d}. Let r be the partial order {(a,a),(b,b), (c,c),(d,d),(b,d),(c,a)}. Give the Hasse diagram for r.
Consider the relation R = {(a,b), (b, b), (b, c), (c, b), (c, c), (d, a), (d, c), (d, d)} on {a, b, c, d}. (a) Draw the digraph for R. (b) Write the adjacency matrix for R with respect to vertices in alphabetical order. (c) Draw the digraph for R o R.
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1. (a) For the invariant theory connected to the general linear model, find g. (b) Show that ğ : 0₁ → 9₁ and 9: 0₂ ā (c) Show that 8² = 0 ₂ for all g. Σ7_₁ ? is the maximal invariant in © = {§1, ..., Ék, 0²}.
Now consider the network (G, w) such that V(G) = {V₁, V2, U3, U4, V5, V6, U7}, E(G) = {V1V2, V1V3, V2V3, V2V4, V2V5, V3V5, V3V6, V4V5, V5V6} and w(v₁v₂) = 1, w(v₂v₁) = 2, w(v3v6) = 1, w (v₁v3) = 4, w(v₂v5) = 3, w(v4v5) = 1, w(v₂V3) = 4 w(v3v5) = 4 w(v5v6) = 3. the spanning three of (GW) is V₂ V6 V5 J 1 2 V4 Show that the spanning tree unique minimum spanning tree of the network (G, w). V/₂ is the
Let V = {V₁, V2, V3} and R = {(V₁, V₁), (V₁, V2), (V₁, V3), (V2, V2), (V2, V3), (V3, V₁)}. a. Draw a directed graph to represent R and find MR. b. Find So, S₁, S2, and the shortest path matrix, S3.
7. Let = [1,2,0]T and = = [1,0,3]¹. (a) (2 points) What is span{u, }? (You can describe this algebraically, geometrically, or in your own words; but be precise!) sparu, √5 - pe { [!] [; }} WA } span =) 44 -2 6 3 3 both they are linearly independent in R3, there fore both vector in form plane PR ³. 01 (b) (3 points) Show that the vector w = [8, 10,9] is in span{ū,7}. C U T Q V =. W 1 8 05 0 U-E-CAD-MA 10 2 16 =) 3 3 9 бо there is no therefore the w is in the yon { U₁U'} piral column in pan (c) (3 points) Is the set {u,, w} linearly independent or linearly dependent? Make sure to justify your answer. from par (6) as he can we could the get at Ilnear dependent from the {0,²}
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3. Consider a network (G, w). Let c € R, and let m: E(G) → R such that m(e) = w(e) + c for all e € E(G). (a) Show that T is a minimum spanning tree of (G, w) if and only if it is a minimum spanning tree of (G,m).